Expanded Form Using Indices

Uganda Primary 7 Mathematics — Topic 3, Term I

How Short Can You Write One Million?

Ordinary form: — seven digits, six zeros

Index form: — three characters

Which would you rather write in a science textbook?

Index Notation — The Vocabulary We Use

• Base: the number being multiplied ()
• Exponent (or index): how many times to multiply ()
• Power: the result ()

We say: "ten to the power three" or "ten cubed"

Powers of 10 — Building the Table

Powers of 10 Meet the Place Value Chart

Every column in the place value chart is a power of 10!

Why Is ?

Follow the pattern (each step divides by 10):

Rule: Any non-zero number to the power zero equals 1.

Chunk 1 Practice — Powers of 10

Evaluate:

4. What power of 10 equals ?
5. Which place value column corresponds to ?

What Is Expanded Form? A Definition

Expanded form writes each digit of a number as:

Number:

Every digit has a term — including zero-value digits.

Worked Example 1 — Expand 5,403

Index form:

Evaluated form:

Worked Example 2 — Expand 80,605

Index form:

Evaluated form:

Two zero-digit terms must be included.

Worked Example 3 — Expand 7,306,024

Index form:

Evaluated form:

The Zero-Digit Term — Why It Matters

If you omit from 5,403:

• Reading coefficients in order: — looks like ✗
• Evaluated sum still gives ✓

Always include every term, including zero ones.

Your Turn — Expand 4,050,300

Write the full expanded form in both index form and evaluated form.

Check: your evaluated form must sum to .

Verify 4,050,300 by Summing Terms

Correct expansion of 4,050,300:

Four zero-digit terms in this number.

Chunk 2 Summary — Expanded Form Rules

Expanded form rules:

1. Write one term per digit column, from highest power to
2. Each term:
3. Include zero-coefficient terms (placeholders)
4. Verify: the evaluated sum must equal the original number

Reverse Direction — Reading Expanded Form

Reconstruct the standard number from its expanded form.

Two strategies:

1. Evaluate and sum: compute each term, then add
2. Read coefficients: write digits high to low power; fill 0 for gaps

Example:

Worked Reconstruction — Rebuild 306,209

Missing powers: and → insert zeros

Coefficients in order:

Tricky Example — Large Gap

Missing powers: — five gaps!

Coefficients:

Misconception 1 — Omitting Zero-Digit Terms

Reading back with a missing zero:

Coefficients read: → someone writes ✗

Fix: always write explicitly.

Misconception 2 — Confusing Exponent with Digit Value

Error: ✗

Why? Digit is 7, but the column is hundreds ().

Fix: exponent = column position (count from right at 0)

Rule: coefficient = digit value; exponent = column position.

Misconception 3 — Thinking

Error: ✗

Fix — use the pattern:

Each step divides by 10. So ✓

Lesson Summary — Two Forms, Same Number

Every whole number has two equivalent forms:

Standard → Expanded: one term per digit, exponent = column power.